Acoustic time-reversal mirrors

Transcription

1 INSTITUTE OF PHYSICS PUBLISHING INVERSE PROBLEMS Inverse Problems 17 (21) R1 R38 PII: S (1) TOPICAL REVIEW Acoustic time-reversal mirrors Mathias Fink and Claire Prada Laboratoire Ondes et Acoustique, Ecole Supérieure de Physique et de Chimie Industrielle de la Ville de Paris, Université Denis Diderot, UMR CNRS 7587, 1 Rue Vauquelin, 755 Paris, France Received 2 August 2 Abstract The objective of this paper is to show that time-reversal invariance can be exploited in acoustics to accurately control wave propagation through random propagating media as well as through waveguides or reverberant cavities. To illustrate these concepts, several experiments are presented. They show that, contrary to long-held beliefs, multiple scattering in random media and multipathing in waveguides and cavities enhances spatial resolution in time-reversal acoustics by making the effective size of time-reversal mirrors much larger than their physical size. Applications of time-reversal mirrors in pulse-echo detection are then described and, for a multi-target medium, the iterative time-reversal mode is presented. It will be shown how the study of the time-reversal operator allows us to select and to focus on each target through a distorting medium or inside a reverberant medium. (Some figures in this article are in colour only in the electronic version; see Introduction The objective of this paper is to show that time-reversal invariance can be exploited in acoustics to accurately control wave propagation and to improve target detection through complex media. In time-reversal acoustics [1 4] a signal is recorded by an array of transducers, timereversed and then re-transmitted into the medium. The re-transmitted signal propagates back through the same medium and refocuses on the source. In a time-reversal cavity (TRC) the array completely surrounds the source and thus the time-reversed signals propagate backwards through the medium and go through all the multiple scattering, reflections and refraction that they underwent in the forward direction. If the time-reversal operation is only performed on a limited angular area (a time-reversal mirror, TRM), a small part of the field radiated by the source is captured and time reversed, thus limiting reversal and focusing quality. The time-reversal approach is clearly connected to the inverse source problem. In both, one deals with propagation of a time-reversed field, but the propagation is real in the TRC experiment and simulated in the inverse problem. The most important distinction, however, is that source localization methods such as back-propagation demand knowledge of the propagating medium, while the TRC method does not /1/11+38$3. 21 IOP Publishing Ltd Printed in the UK R1

2 R2 Topical review Different types of time-reversal experiment described in this paper show that focusing quality is improved if the wave traverses random media or if the wave propagates in media with reflecting boundaries as waveguides or reverberating cavities. The focusing resolution may be much better than the resolution obtained in a homogeneous medium. Multiple scattering or multiple reflections allow us to redirect one part of the initial wave towards the TRM, that normally misses the transducer array. Roughly speaking, the inhomogeneities or the boundaries produce multi-pathing and the TRM appears to have an aperture that is much larger than its physical size. TRM properties are not only limited to focusing a wave on an active source. They can also be used in multi-target detection through complex media. Combined with pulse-echo detection, the iterative time-reversal mode allows us to focus on the most reflective target. It will be shown how an extension of the time-reversal technique, the DORT (Diagonalisation de l Operateur de Retournement Temporel) method, allows us to focus individually on each target. Improvement of the resolution is also observed in reverberating media with the DORT method. 1. Time-reversal cavities and mirrors The basic theory employs a scalar wave formulation φ( r,t)and, hence, is strictly applicable to acoustic or ultrasound propagations in fluid. However, the basic ingredients and conclusions apply equally well to elastic waves in a solid and to electromagnetic fields. In any propagation experiment, the acoustic sources and the boundary conditions determine a unique solution φ( r,t) in the fluid. The goal, in time-reversal experiments, is to modify the initial conditions in order to generate the dual solution φ( r,t t) where T is a delay due to causality requirements. Jackson and Dowling and Cassereau and Fink [4, 5] have studied theoretically the conditions necessary to ensure the generation of φ( r,t t) in the entire volume of interest The time-reversal cavity From an experimental point of view a TRC consists of a two-dimensional piezoelectric transducer array that samples the wavefield over a closed surface. An array pitch of the order of λ/2, where λ is the smallest wavelength of the pressure field, is needed to ensure the recording of all the information on the wavefield. Each transducer is connected to its own electronic circuitry that consists of a receiving amplifier, an A/D converter, a storage memory and a programmable transmitter able to synthesize a time-reversed version of the stored signal. Although reversible acoustic retinas usually consist of discrete elements, it is convenient to examine the behaviour of idealized continuous retinas, defined by two-dimensional surfaces. In the case of a TRC, we assume that the retina completely surrounds the source. The basic time-reversal experiment can be described in the following way. In a first step, a point-like source located at r, inside a volume V surrounded by the retina surface S, emits a pulse at t = t. The wave equation in a medium of density ρ( r) and compressibility κ( r) is given by (L r + L t )φ( r,t) = Aδ( r r )δ(t t ) ( ) 1 L r = ρ( r) (1) L t = κ( r) tt where A is a dimensional constant that ensures the compatibility of physical units between the two sides of the equation; for simplicity reasons, this constant will be omitted in the

3 Topical review R3 following. The solution to equation (1) reduces to the Green function G( r,t r,t ). Classically, G( r,t r,t ) is written as a diverging spherical wave (homogeneous and free-space case) and additional terms that describe the interaction of the field itself with the inhomogeneities (multiple scattering) and the boundaries. We assume that we are able to measure the pressure field and its normal derivative at any point on the surface S during the interval [,T]. As time-reversal experiments are based on a two-step process, the measurement step must be limited in time by a parameter T. In all the following, we suppose that the contribution of multiple scattering decreases with time, and that T is chosen such that the information loss can be considered as negligible inside the volume V. During the second step of the time-reversal process, the initial source at r is removed and we create on the surface of the cavity monopole and dipole sources that correspond to the time reversal of these same components measured during the first step. The time-reversal operation is described by the transform t T t and the secondary sources are φ s ( r,t) = G( r,t t r,t ) (2) n φ s ( r,t) = n G( r,t t r,t ). In this equation, n is the normal derivative operator with respect to the normal direction n to S, oriented outward. Due to these secondary sources on S, a time-reversed pressure field φ tr ( r 1,t 1 ) propagates inside the cavity. It can be calculated using a modified version of the Helmoltz Kirchhoff integral φ tr ( r 1,t 1 ) = + dt S [G( r 1,t 1 r,t) n φ s ( r,t) φ s ( r,t) n G( r 1,t 1 r,t)] d2 r ρ( r). (3) Spatial reciprocity and time-reversal invariance of the wave equation (1) yield the following expression: φ tr ( r 1,t 1 ) = G( r 1,T t 1 r,t ) G( r 1,t 1 r,t t ). (4) This equation can be interpreted as the superposition of incoming and outgoing spherical waves, centred on the initial source position. The incoming wave collapses at the origin and is always followed by a diverging wave. Thus the time-reversed field, observed as a function of time, from any location in the cavity, shows two wavefronts, where the second one is the exact replica of the first one, multiplied by 1. If we assume that the retina does not perturb the propagation of the field (free-space assumption) and that the acoustic field propagates in a homogeneous fluid, the free-space Green function G reduces to a diverging spherical impulse wave that propagates with a sound speed c. Introducing its expression in (4) yields the following formulation of the time-reversed field: φ tr ( r 1,t 1 ) = K( r 1 r,t 1 T + t ) (5) where the kernel distribution K( r,t) is given by ( t + r c ) 1 4π r δ ( t r ). (6) c K( r,t) = 1 4π r δ The kernel distribution K( r,t) corresponds to the difference between two impulse spherical waves that respectively converge to and diverge from the origin of the spatial coordinate system, thus the location of the initial source. It results from this superposition that the pressure field remains finite for all time throughout the cavity, although the converging and diverging spherical waves show a singularity at the origin. Note that this singularity occurs at time t 1 = T t.

4 R4 Topical review The time-reversed pressure field, observed as a function of time, shows two wavefronts, where the second one is the exact replica of the first one, multiplied by 1. If we consider a wide-band excitation function instead of a Dirac distribution δ(t), the two wavefronts overlap near the focal point, therefore resulting in a temporal distortion of the acoustic signal. It can be shown that this distortion yields a temporal derivation of the initial excitation function at the focal point. If we now calculate the Fourier transform of (6) over the time variable t, we obtain K( r,ω) = 1 sin(ω r /c) = 1 sin(k r ) (7) 2jπ r jλ k r where λ and k are the wavelength and wavenumber, respectively. As a consequence, the time-reversal process results in a pressure field that is effectively focused on the initial source position, but with a focal spot size limited to one half-wavelength. The size of the focal spot is a direct consequence of the superposition of the two wavefronts and can be interpreted in terms of the diffraction limitations (loss of the evanescent components of the acoustic fields). A similar interpretation can be given in the case of an inhomogeneous fluid, but the Green function G now takes into account the interaction of the pressure field with the inhomogeneities of the medium. If we were able to create a film of the propagation of the acoustic field during the first step of the process, the final result could be interpreted as a projection of this film in the reverse order, immediately followed by a re-projection in the initial order. The apparent failure of the time-reversed operation that leads to diffraction limitation can be interpreted in the following way: the second step described above is not strictly the time reversal of the first step: during the second step of an ideal time-reversed experiment, the initial active source (that injects some energy into the system) must be replaced by a sink (the time reversal of a source). An acoustic sink is a device that absorbs all arriving energy without reflecting it. de Rosny, using the source as a diverging wavefront canceller, has recently built such a sink in our laboratory, and he has observed a focal spot size significantly below diffraction limits [6] The time-reversal mirror This theoretical model of the closed TRC is interesting since it affords an understanding of the basic limitations of the time-reversed self-focusing process; but it has some limitations, particularly compared to an experimental setup. It can be proven that it is not necessary to measure and time reverse both the scalar field (acoustic pressure) and its normal derivative on the cavity surface: measuring the pressure field and re-emitting the time-reversed field in the backward direction yields the same results, on the condition that the evanescent parts of the acoustic fields have vanished (propagation along several wavelengths) [7]. This arises from the fact that each transducer element of the cavity records the incoming field from the forward direction, and retransmits it (after the time-reversal operation) in the backward direction (and not in the forward direction). The change between the forward and the backward direction replaces the measurement and the time reversal of the field normal derivative. From an experimental point of view, it is not possible to measure and re-emit the pressure field at any point of a 2D surface: experiments are carried out with transducer arrays that spatially sample the receiving and emitting surface. The spatial sampling of the TRC by a set of transducers may introduce grating lobes. These lobes can be avoided by using an array pitch smaller than λ min /2, where λ min is the smallest wavelength of the transient pressure field. In this latest case, each transducer senses all the wavevectors of the incident field.

5 Topical review R5 source array (first step) hydrophone xaxis Figure 1. Sketch of the time-reversal experiment through a random medium. The temporal sampling of the data recorded and transmitted by the TRC has to be at least of the order of T min /8 (T min minimum period) to avoid secondary lobes [8]. It is generally difficult to use acoustic arrays that surround completely the area of interest, and the closed cavity is usually replaced by a TRM of limited angular aperture. This yields an increase of the point spread function dimension that is usually related to the mirror angular aperture observed from the source. 2. Time-reversal experiments 2.1. Time-reversal through random media Derode et al [9] carried out the first experimental demonstration of the reversibility of an acoustic wave propagating through a random collection of scatterers with strong multiplescattering contributions. In an experiment such as that depicted in figure 1, a multiple-scattering sample is placed between the source and an array made of 128 elements. The whole set-up is in a water tank. The scattering medium consists of a set of 2 parallel steel rods (diameter.8 mm) randomly distributed. The sample thickness is L = 4 mm, and the average distance between rods is 2.3 mm. The source is 3 cm away from the TRM and transmits a short (1 µs) ultrasonic pulse (three cycles of 3.5 MHz, figure 2(a)). Figure 2(b) shows the waveform received on the TRM by one of the elements. It is spread over 25 µs, i.e. 25 times the initial pulse duration. A long incoherent wave is observed, which results from the multiply scattered contribution. In the second step of the experiment, the 128 signals are time reversed and transmitted and a hydrophone measures the time-reversed wave around the source location. Two different aspects of this problem have been studied: the property of the signal recreated at the source location (time compression) and the spatial property of the time-reversed wave around the source location (spatial focusing). The time-reversed wave traverses the rods back to the source, and the signal received

6 R6 Topical review a) b) c) Amplitude Amplitude Amplitude Signal transmitted in water and received on transducteur Time (µs) Signal transmitted through the multiple scattering sample and received on transducteur Time (µs) Signal received on the source Time (µs) Figure 2. (a) Signal transmitted by the source, (b) signal recorded by an array element, (c) timereversed signal observed at the source location. at the source is represented in figure 2(c): an impressive compression is observed, since the received signal lasts about 1 µs, against 25 µs. The pressure field is also measured around the source, in order to obtain the directivity pattern of the beam emerging from the rods after time reversal, and the results are plotted in figure 3. Surprisingly, multiple scattering has not degraded the resolution of the system: indeed, the resolution is found to be six times finer (bold curve) than the classical diffraction limit (thin curve). This effect does not contradict the laws of diffraction, though. The intersection of the incoming wavefront with the sample has a typical size D. After time reversal, the waves travel on the same scattering paths and focus back on the source as if they were passing through a converging lens with size D. The angular aperture of this pseudo-lens is much wider than that of the array alone, hence the improvement in resolution. In other words, because of the scattering sample, the array is able to detect higher spatial frequencies than in a purely homogeneous medium. High spatial frequencies that would have been lost otherwise are redirected towards the array, due to the presence of the scatterers in a large area.

7 Topical review R db -2-3 mm Figure 3. Patterns of the time-reversed fields in the source plane through the random medium (bold curve) and through a homogeneous fluid (thin curve). This experiment shows also that the acoustic time-reversal experiments are surprisingly stable. The recorded signals have been sampled with 8 bit analogue-to-digital converters that introduce quantization errors and the focusing process still works. This has to be compared with time-reversal experiments involving particles moving like balls on an elastic billiard of the same geometry. Computation of the direct and reversed particle trajectory moving in a plane among a fixed array of some thousand obstacles shows that the complete trajectory is irreversible. Indeed, such a system is a well known example of a chaotic system that is highly sensitive to initial conditions. The finite precision that occurs in the computer leads to an error in the trajectory of the time-reversed particle that grows exponentially with the number of scattering encounters. Recently, Snieder and Scales [1] have performed numerical simulations to point out the fundamental difference between waves and particles in the presence of multiple scattering by random scatterers. In fact, they used time reversal as a diagnostic of wave and particle chaos: in a time-reversal experiment, a complete focusing on the source will only take place if the velocity and positions are known exactly. The degree δ to which errors in these quantities destroy the quality of focusing is a diagnostic of the stability of the wave or particle propagation. Intuitively, the consequences of a slight deviation δ in the trajectory of a billiard ball will become more and more obvious as time goes on, and as the ball undergoes more and more collisions. Waves are much less sensitive than particles to initial conditions. Precisely, in a multiple-scattering situation, the critical length scale δ that causes a signification deviation at a time t in the future decreases exponentially with time in the case of particles, whereas it only decreases as the square root of time for waves in the same situation. Waves and particles react in fundamentally different ways to perturbations of the initial conditions. The physical reason for this is that each particle follows a well defined trajectory whereas waves travel along all possible trajectories, visiting all the scatterers in all possible combinations. While a small error in the initial velocity or position makes the particle miss one obstacle and completely change its future trajectory, the wave amplitude is much more stable because it results from the interference of all the possible trajectories and small errors in the transducer operations will sum up in a linear way, resulting in small perturbation Time reversal as time correlator. Like any linear and time-invariant process, wave propagation through a multiple-scattering medium may be described as a linear system with

8 R8 Topical review different impulse responses. If a source, located at r, sends a Dirac pulse δ(t), the jth transducer of the TRM will record the impulse response h j (t) that corresponds, for a point transducer located at r j, to the Green function G( r j,t r, ). Moreover, due to reciprocity, h j (t) is also the impulse response describing the propagation of a pulse from the jth transducer to the source. Thus, neglecting the causal time delay T, the time-reversed signal at the source is equal to the convolution product h j (t) h j ( t). This convolution product, in terms of signal analysis, is typical of a matched filter. Given a signal as input, a matched filter is a linear filter whose output is optimal in some sense. Whatever the impulse response h j (t), the convolution h j (t) h j ( t) is maximum at time t =. This maximum is always positive and is equal to h 2 j (t) dt, i.e. the energy of the signal h j (t). This has an important consequence. Indeed, with an N-element array, the time-reversed signal recreated on the source is written as a sum j=n φ tr ( r,t)= h j (t) h j ( t). (8) j=1 Even if the h j (t) are completely random and apparently uncorrelated signals, each term in this sum reaches its maximum at time t =, so all contributions add constructively around t =, whereas at earlier or later times uncorrelated contributions tend to destroy one another. Thus the re-creation of a sharp peak after time reversal on an N-element array can be viewed as an interference process between the N outputs of N matched filters. The robustness of the TRM can also be accounted for through the matched filter approach. If, for some reason, the TRM does not exactly retransmit h j ( t) but rather h j ( t) + n j (t), where n j (t) is additional noise on channel j, then the re-created signal is written j=n j=1 j=n h j (t) h j ( t) + j=1 h j (t) n(t). The time-reversed signals h j ( t) are tailored to exactly match the medium impulse response, which results in a sharp peak, whereas an additional small noise is not matched to the medium and, given the extremely long duration involved, it generates a low-level long-lasting background noise instead of a sharp peak Time reversal as a spatial correlator. Another way to consider the focusing properties of the time-reversed wave is to follow the impulse response approach and treat the time-reversal process as a spatial correlator. If we denote by h j (t) the propagation impulse response from the jth element of the array to an observation point r 1 different from the source location r, the signal recreated in r 1 at time t 1 = is written φj tr ( r 1, ) = h j (t)h j (t) dt. (9) Notice that this expression can be used as a way to define the directivity pattern of the timereversed waves around the source. Now, due to reciprocity, the source S and the receiver can be exchanged, i.e. h j (t) is also the signal that would be received in r 1 if the source were the jth element of the array. Therefore, we can imagine this array element is the source, and the transmitted field is observed at two points r 1 and r. The spatial correlation function of this wavefield would be h j (t)h j (t), where the impulse response product is averaged on different realizations of the disorder. Therefore equation (9) can be viewed as an estimator of this spatial correlation function. Note that in one time-reversal experiment we have only access to a single realization of the disorder. However, the ensemble average can be replaced by a time average, a frequency average or a spatial average on a set of transducers.

9 Topical review R db Distance from the source (mm) Figure 4. Directivity patterns with N = 122 transducers (bottom curve) and N = 1 transducer (top curve). In that sense, the spatial resolution of the TRM (i.e. the 6 dbwidth of the directivity pattern) is simply an estimate of the correlation length of the scattered wavefield [11]. This has an important consequence. Indeed, if the resolution of the system essentially depends on correlation properties of the scattered wavefield, it should become independent from the array s aperture. This is confirmed by the experimental results. Figure 4 presents the directivity patterns obtained through a 4 mm thick multiple-scattering sample, using either one array element or the whole array (122 elements) as a TRM. In both cases, the spatial resolution at 6 db is the same:.85 mm. In contradiction with what happens in a homogeneous medium, enlarging the aperture of the array does not change the 6 db spatial resolution. However, even though the number N of active array elements does not influence the typical width of the focal spot, it has a strong impact on the background level of the directivity pattern ( 12 db for N = 1, 28 db for N = 122), as can be seen in figure 4. Finally, the fundamental properties of time reversal in a random medium rely on the fact that it is both a space and time correlator, and the time-reversed waves can be viewed as an estimate of the space and time auto-correlation functions of the waves scattered by a random medium. The estimate becomes better with a large number of transducers in the mirror. Moreover, the system is not sensitive to a small perturbation since adding a small noise to the scattered signals (e.g. by digitizing them on a reduced number of bits) may alter the noise level but does not drastically change the correlation time or the correlation length of the scattered waves. Even in the extreme case where the scattered signals are digitized on a single bit, Derode has shown recently [12] that the time and space resolution of the TRM are practically unchanged, which is striking evidence for the robustness of wave time reversal in a random medium Time reversal in bounded media In the TRC approach, the transducer array samples a closed surface surrounding the acoustic source. In the last paragraph, we have seen how the multiple-scattering processes in a large sample widen the effective TRM aperture. The same kind of improvement may be obtained for waves propagating in a waveguide or in a cavity. Multiple reflections along the medium

10 R1 Topical review boundaries significantly increase the apparent aperture of the TRM. The basic idea is to replace some of the TRC transducers by reflecting boundaries that redirect one part of the incident wave towards the TRM aperture. Thus spatial information is converted into the time domain and the reversal quality depends crucially on the duration of the time-reversal window, i.e. the length of the recording to be reversed Ultrasonic waveguide. Experiments conducted by Roux in rectangular ultrasonic waveguides have shown the effectiveness of the TR processing to compensate for multipath effects [13]. The experiment is conducted in a water waveguide whose interfaces are plane and parallel. The length of the guide is L 8 mm along the y-axis, with a vertical water depth H 4 mm along the x-axis. A point-like ultrasonic source is located on one side of the waveguide. On the other side, a TRM, identical to that used in the multiple-scattering medium, is used. In total, 96 of the array elements are used, which corresponds to an array aperture equal to the waveguide aperture. Figure 5(a) shows the field radiated by the source and recorded by the transducer array after propagation through the channel. After the arrival of the first wavefront corresponding to the direct path we observe a set of signals, due to multiple reflections of the incident wave between the interfaces, that spread over 1 µs. Figure 5(b) represents the signal received on one transducer of the TRM. After the time-reversal operation of the 1 µs signals, we observe the spatio-temporal distribution of the time-reversed field on the source plane (figure 6(a)) and we note a remarkable temporal compression at the source location (figure 6(b)). This means that multipath effects are fully compensated. The signal observed at the source is nearly identical to that received in a time-reversed experiment conducted in free space. In this experiment, the transfer function of the waveguide has been completely compensated by the time-reversal process. As with a multiple-scattering medium, the timereversal process enables the realization of an optimal matched filter of the waveguide transfer function. The analysis of figure 6 shows that the ratio between the peak signal and the side lobe level is of the order of 45 db. Figure 7 shows the directivity pattern of the time-reversed field observed in the source plane. The time-reversed field is focused on a spot which is much smaller than that obtained with the same TRM in free space. In our experiment, the 6 db lateral resolution is improved by a factor of nine. This can be easily interpreted by the image theorem in a medium bounded by two mirrors. For an observer located at the source point, the 4 mm TRM appears to be accompanied by a set of virtual images related to multipath reverberation. The effective TRM is then a set of TRMs as shown in figure 8. When taking into account the first ten arrivals, the theoretical effective aperture of the mirror array is ten times larger than the real aperture. However, in practice, as the replicas arrive later, their amplitudes decrease. The angular directivity of the transducers leads to an apodization of the effective aperture of the TRM Application of time-reversal processing in underwater acoustics. Acoustic waveguides are currently found in underwater acoustics, especially in shallow water, where multipath propagation limits the capacity of underwater communication systems. The problem arises because acoustic transmission in shallow water bounces off the ocean surface and floor, so a transmitted pulse gives rise to multiple arrivals at the receiver. To compensate for acoustic reverberation in the ocean, single-channel time reversal was first introduced in the early 196s by Parvulescu and Clay [14, 15]. They performed

11 Topical review R11 Depth in the waveguide (mm) (a) Time ( µs) Amplitude (b) Time ( µs) Figure 5. (a) Signals recorded, through the waveguide, by the array elements (vertical axis) in a B scan format; (b) signal recorded by one array element. experiments in shallow water at sea with one transducer working in a time-reversed mode. They observed temporal compression but their experiments did not embody the spatial focusing property of TRMs. Parvulescu s approach consists of considering the ocean as a correlator. Jackson and Dowling [4] developed a theoretical formalism to describe phase conjugation in the ocean. This formalism is based on the modal decomposition of the pressure field in an acoustic waveguide. Following this approach, Feuillade and Clay [16] computed in 1992 numerical time-reversal experiments in shallow water. Since 1996, Kuperman et al [17 19] have performed several underwater acoustics experiments in a 12 m deep ocean waveguide. At frequencies of 5 Hz and 3.5 khz, they used a 24-element TRM to accomplish time-reversal focusing and multipath compensation from 7 km up to 3 km. Theoretically speaking, for one spectral component at frequency ω, the time-reversal operation consists of a phase conjugation of the incident field. For an incident field from a point source located at depth x s in the plane y = L, the phase conjugation, performed in shallow water, from a vertical array of N discrete sources located in the plane y = leads to

12 R12 Topical review Depth in the waveguide (mm) (a) Time (µs) 1 Amplitude (b),5 -, ,5 2,5 5 Time (µs) Figure 6. (a) Time-reversed field recorded in the source plane (vertical axis) versus time (horizontal axis); (b) zoom of the time-reversed signal recorded at the source location through the waveguide and without. Amplitude (db) mm Figure 7. Patterns of the time-reversed fields through the waveguide (full curve) and in an open space (dotted curve).

13 Topical review R13 point source S TRM image real TRM aperture of the TRM in free water aperture of the TRM in the waveguide Figure 8. Virtual images in a waveguide. For an observer located at the source position, a TRM appears to be surrounded by a set of virtual images that increase its apparent aperture. the time-reversed pressure field at observation point (x, y): φ tr (x,y,t)= N G ω (x, y x j, )G ω (x j, x s,l)exp( iωt) (1) j=1 where G ω (x j, x s,l)is the conjugated monochromatic Green function of the waveguide at frequency ω between a source at depth x s and range L and a receiver at depth x j and range. In other words, the phase-conjugated field in the plane of the source is the sum over the array elements of a product of two Green functions: one describes the propagation from the source to the array and the other describes the propagation from the array to the observation plane. Time reversal appears in the conjugation of the Green function between the source and the array in the right-hand term of equation (1). In a range-independent waveguide, the Green function is expressed as follows: G ω (x, x s,l)= i ρ(x s )(8πL) 1/2 exp ( i π 4 ) n u n (x s )u n (x) k 1/2 n exp(ik n L) (11) where n is the number of propagating modes, u n (x) corresponds to the modal shape as a function of depth and k n is the wavenumber. To demonstrate that φ tr (x,y,t) focuses at the position of the initial source, we simply substitute equation (11) into (1), which specifies that we sum over all modes and array sources: φ tr (x,y,t) u m (x)u m (x j )u n (x j )u n (x s ) j n m ρ(x j )ρ(x s ) k m k n yl exp(i(k my k n L) ωt). (12) For an array which substantially spans the water column, we approximate the sum of sources as an integral and invoke the orthonormality of the modes: u m (x)u n (x) dx = δ nm. (13) ρ(x) The sum over j selects mode m = n and equation (12) becomes φ tr (x,y,t) u m (x)u m (x s ) exp(ik m (y L) ωt). (14) m ρ(x s )k m yl In the plane of ( the source at y = L, the closure relations which define the modes ) u as a complete set m (x)u m (x s ) m ρ(x s = δ(x x ) s ) can be applied under the assumption that the k n are nearly constant over the interval of the contributing modes. This leads to

14 R14 Topical review φ tr (x,l,t) δ(x x s ) exp( iωt), which proves that the phase conjugated field focuses back at the source. Kuperman et al have experimentally demonstrated in the ocean the robustness of timereversal focusing provided the array adequately samples the field in the water column. They have shown that temporal changes in the ocean due to surface waves and internal waves degrade the focus but that this degradation is tolerable if the average Green function is not severely perturbed by these time variations [17]. Moreover they experimentally achieved a shift of the focal range on the order of 1% by shifting the central frequency of the TRM prior to retransmission [18] Time reversal in chaotic cavities In this section, we are interested in another aspect of multiply reflected waves: waves confined in closed reflecting cavities such as elastic waves propagating in a silicon wafer. With such boundary conditions, no information can escape from the system and a reverberant acoustic field is created. If, moreover, the cavity shows ergodic properties and negligible absorption, one may hope to collect all information at only one point. Draeger and Fink [19 21] have shown experimentally and theoretically that in this particular case a time reversal can be obtained using only one TR channel operating in a closed cavity. The field is measured at one point over a long period of time and the time-reversed signal is re-emitted at the same position. The experiment is two dimensional and has been carried out by using elastic surface waves propagating along a monocrystalline silicon wafer whose shape is a chaotic stadium. The shape of the cavity is of crucial importance. The chaotic stadium geometry ensures that each acoustic ray radiated by the source will pass, after several reflections, sufficiently close to any point of the cavity. This ergodic property may be obtained for different geometries, and the selected geometry, called a D-shape stadium, was chosen for its simplicity. Silicon was selected for its weak absorption. The elastic waves which propagate in such a plate are Lamb waves. An aluminium cone coupled to a longitudinal transducer generates these waves at one point of the cavity. A second transducer is used as a receiver. The central frequency of the transducers is 1 MHz and its bandwidth is 1%. At this frequency, only three Lamb modes are possible (one flexural, two extensional). The source is isotropic and considered to be point-like because the cone tip is much smaller than the central wavelength. A heterodyne laser interferometer measures the displacement field as a function of time at different points on the cavity. Assuming that there is hardly any mode conversion between the flexural mode and other modes at the boundaries, we have only to deal with one field, the flexural-scalar field. The experiment is a two-step process as described above: in the first step, one of the transducers, located at point A, transmits a short omnidirectional signal of duration.5 µs into the wafer. Another transducer, located at B, observes a very long chaotic signal, that results from multiple reflections of the incident pulse along the edges of the cavity, and which continues for more than 5 ms, corresponding to some hundred reflections along the boundaries. Then, a portion of 2 ms of the signal is selected, time reversed and re-emitted by point B. As the timereversed wave is a flexural wave that induces vertical displacements of the silicon surface, it can be observed using the optical interferometer that scans the surface around point A (figure 9). One observes both an impressive time recompression at point A and a refocusing of the time-reversed wave around the origin (figures 1(a) and (b)), with a focal spot whose radial dimension is equal to half the wavelength of the flexural wave. Using reflections at the boundaries, the time-reversed wavefield converges towards the origin from all directions and gives a circular spot, like the one that could be obtained with a closed TRC covered with

15 Topical review R15 A B ~ 2ms? reversal A B Figure 9. In the first step, a point transducer located at A transmits a flexural wave into the silicon wafer that is recorded at point B. Then a portion of the recorded signal is time reversed and retransmitted by point B. transducers. The 2 ms time-reversed waveform is the time sequence needed to focus exactly on point A. The success of this time-reversal experiment is particularly interesting with respect to two aspects. Firstly, it proves again the feasibility of time reversal in wave systems with chaotic ray dynamics. Paradoxically, in the case of one-channel time reversal, chaotic dynamics is not only harmless but also even useful, as it guarantees ergodicity. Secondly, using a source of vanishing aperture, we obtain an almost perfect focusing quality. The procedure approaches the performance of a closed TRC, which has an aperture of 36. Hence, a one-point time reversal in a chaotic cavity produces better results than a TRM in an open system. Using reflections at the edge, focusing quality is not aperture limited and, in addition, the time-reversed collapsing wavefront approaches the focal spot from all directions. As for the multiple-scattering medium, focusing properties of the time-reversed wave can be calculated using the spatial correlator approach. Taking into account the modal decomposition of the impulse response h AB (t) on the eigenmodes ψ n ( x) of the cavity with eigenfrequency ω n, we obtain h AB (t) = ψ n (A)ψ n (B) sin(ω nt) (t > ). (15) n ω n If we denote by h A B(t) the propagation impulse response from point B to an observation point A (with coordinates r 1 ) different from the source location A, the time-reversed signal recreated at A at time t 1 = is written φ tr ( r 1, ) = h AB (t)h A B(t) dt. (16) Thus the directivity pattern of the time-reversed wavefield is given by the cross correlation of

16 R16 Topical review (a) (b) wafer scanned region 15 mm 15 mm (a) R (b) R (c) R (d) R (e) R (f) R Figure 1. (a) Time-reversed signal observed at point A. The observed signal is 21 µs long. (b) Time-reversed wavefield observed at different times around point A on a square of mm 2. the Green functions that can be developed on the eigenmodes of the cavity φ tr ( r 1, ) = 1 ψ n ωn 2 n (A)ψ n ( r 1 )ψn 2 (B). (17) Note that in a real experiment one has to take into account the limited bandwidth of the transducers, so a spectral function F(ω) centred on frequency ω c, with bandwidth ω, must be introduced and we can write equation (17) in the form φ tr ( r 1, ) = 1 ψ n ωn 2 n (A)ψ n ( r 1 )ψn 2 (B)F (ω n). (18) Thus the summation is limited to a finite number of modes, that is typically in our experiment of the order of some hundreds. As we do not know the exact eigenmode distribution for each chaotic cavity, we cannot evaluate this expression directly. However, one may use a statistical approach and consider the average over different realizations, which consists in summing over

17 Topical review R17 different cavity realizations. So we replace in equation (18) the eigenmode product by their expectation values. We use also a qualitative argument proposed by Berry [22 24] to characterize irregular modes in chaotic system. If chaotic rays support an irregular mode, it can be considered as a superposition of a large number of plane waves with random direction and phase. This implies that the amplitude of an eigenmode has a Gaussian distribution with ψn 2 =σ2 and a short-range isotropic correlation function given by a Bessel function that is written ψ n (A)ψ n ( r 1 ) =J (2π r 1 r /λ n ) (19) with λ n is the wavelength corresponding to ω n. If A and A are sufficiently far apart from B not to be correlated, then ψ n (A)ψ n ( r 1 )ψn 2 (B) = ψ n(a)ψ n ( r 1 ) ψn 2 (B). (2) One obtains finally φ tr ( r 1, ) = n 1 J (2π r 1 r /λ n )σ 2 F(ω n ). (21) ω 2 n The experimental results obtained in figure 1 agree with this prediction and show that in a chaotic cavity the spatial resolution is independent of the TRM aperture. Indeed, with a onechannel TRM, the directivity pattern at t = is closed to the Bessel function J (2π r 1 r /λ c ) corresponding to the central frequency of the transducers. One can also observe that a very good estimate of the eigenmode correlation function is experimentally obtained with only one realization. A one-channel omnidirectional transducer is able to refocus a wave in a chaotic cavity, and we have not averaged the data on different cavities or on different positions of the transducer B Phase conjugation versus time reversal. This interesting result emphasizes the great interest of time-reversal experiments, compared to phase-conjugated experiments. In phase conjugation, one works only with monochromatic waves and not with broadband pulses. For example, if one works only at frequency ω n, so that there is only one term in equation (18), one cannot refocus a wave on point A. An omnidirectional transducer, located at any position B, working in monochromatic mode, sends a diverging wave in the cavity that has no reason to refocus on point A. The refocusing process works only with broadband pulses, with a large number of eigenmodes in the transducer bandwidth. Here, the averaging process that gives a good estimate of the spatial correlation function is not obtained by summing over different realizations of the cavity, as in equation (18), but by a sum over pseudo-realizations, which corresponds to the different modes in the same cavity. This arises from the fact that, in a chaotic cavity, we may assume a statistical decorrelation of the different eigenmodes. As the number of eigenmodes available in the transducer bandwidth increases, the refocusing quality becomes better and the focal spot pattern becomes closed to the ideal Bessel function. Hence, the signal to noise level should increase as the square root of the number of modes in the transducer bandwidth. A similar result has also been observed in the time-reversal experiment conducted in multiply scattering medium. A clear refocusing has been obtained with only a single array element (figure 4). The focusing process works with broadband pulses (the transducer centre frequency is 3.5 MHz with a 5% bandwidth at 6 db). For each individual frequency there is no focusing and the estimate of the spatial correlation is very noisy. However, for a large bandwidth, if we have statistical decorrelation of the wavefields for different frequencies, the time-reversed field is self-averaging.

18 R18 Topical review 3. The time-reversal mirror in pulse-echo mode 3.1. Introduction One of the most promising areas for the application of TRMs is pulse-echo detection. In this domain, one is interested in detection, imaging and even destruction of passive reflecting targets. In general, the acoustic detection efficiency depends on the ability to focus a beam in the medium of interest. The presence of sound speed fluctuations between the targets and the transducers can drastically change the beam profiles. In nondestructive evaluation, cracks and defects can be found within materials of various shapes using a coupling liquid; the interface between the samples and the liquid currently degrades the focusing, rendering the detectability of small defects more difficult. In medical applications, one looks for organ walls, calcification or kidney stones, and a fat layer of varying thickness, bone tissues or some muscular tissues may greatly degrade focusing. In underwater acoustics, the objects to detect can be submarines, or mines eventually buried under sediments. In this case, refraction due to oceanic structure ranging in scale from centimetres to tens of kilometres is an important source of distortions. For all these applications, a TRM array can be controlled according to a three-step sequence [25]. One part of the array generates a brief pulse to illuminate the region of interest through the distorting medium. If the region contains a point reflector, the reflected wavefront is selected by means of a temporal window and then the acquired information is time reversed and re-emitted. The re-emitted wavefront refocuses on the target through the medium. It compensates also for unknown deformation of the mirror array. As has been shown the TR processing is a realization of the spatiotemporally matched filter (see section 2.1) to the propagation transfer between the array and the target The principle. Although the time-reversal self-focusing technique is highly effective, it requires the presence of a reflecting target in the medium. When this medium contains several targets, the problem is more complicated and iteration of the TR operation may be used to select one target. Indeed, if the medium contains two targets of different reflectivity, the time reversal of the echoes reflected from these targets generates two wavefronts focused on each target. The mirror produces the real acoustic images of the two reflectors on themselves. The highest-amplitude wavefront illuminates the most reflective target, while the weakest wavefront illuminates the second target. In this case, the time-reversal process can be iterated. After the first time-reversed illumination, the weakest target is illuminated more weakly and reflects a fainter wavefront than that from the strongest target. After some iterations, the process converges and produces a wavefront focused on the most reflective target if the target separation is sufficient to avoid the illumination of one target by the real acoustic image of the other one [26]. Although this simple presentation implies the iterative mode is very attractive, one can argue that a contradiction exists between the concept of iteration and the physical principle of time-reversal invariance. Indeed, the complete time reversal of an acoustic scene results in the time-reversed scene. Therefore, the iteration of the time-reversal operation gives stationary results, a contradiction with wavefield modification after each iteration. In fact, the contradiction is only apparent. A complete time-reversal operation requires a closed TRC surrounding the acoustic scene and a recording time T long enough to take into account all the multiple-scattered waves. In our technique, we utilize only a finite spatial aperture and short temporal windows. Therefore, some information is lost. This information loss gives the iterative mode its target selection capabilities.

19 Topical review R19 Array of transducers. wire 1 wire 2 Needle probe Figure 11. Experimental set-up Experimental validation. Experiments have been performed with a linear array of 128 transducers of central frequency 3.5 MHz similar to those used in section 2. The scatterers were.2 mm and.4 mm diameter copper wires separated by 1 mm and placed at 9 mm distance perpendicular to the array. A rubber aberrating layer was placed between the array and the two wires (figure 11). In the first step, the beam was transmitted by a single transducer element located at the array centre. After the first illumination, the echoes from the two targets were recorded (figure 12(a)). The recorded signals were then time reversed and retransmitted and this process iterated four times. The signals recorded after the fourth iteration are shown in figure 12(b). One of the wavefronts of figure 12(a) was selected; the wavefront has a longer duration because of the multiple iterations. At each iteration, the time-reversed wave was measured 9 mm from the array with a hydrophone. Figure 13 represents the pressure field, which shows two maxima corresponding to the two target locations. The pressure field reaches a higher value at the location of the wire of larger scattering cross section. As the process was iterated the weaker wire received a weaker time-reversed wave. The smallest wire is no longer illuminated after the fourth illumination. This iterative process has also been studied for applications like flaw detection [27] in order to obtain better signal to noise ratio. It was also applied to kidney stone destruction [28]. In this case, the process is used to track, in real time, the dominant bright point of a moving stone. In general, the iterative time-reversal process is not trivial and some elements of theory are now required to better understand the convergence of this process Theory of the iterative time-reversal process In this section, the convergence of TR iterations in a multiple-target medium is studied and conditions for selecting the most reflective targets are derived. The first step is to describe one transmit receive operation in a general manner by defining the transfer matrix of the system The transfer matrix. We consider the array of transducers with all its electronic devices insonifying a given scattering medium. As for conventional ultrasonic imaging devices, the system is supposed to be linear. It is also assumed to be time invariant during the iterative time-reversal process. Such a system can be described by its impulse response function. If

20 R2 Topical review time (µs) transducers time (µs) transducers Figure 12. (a) Echographic signals observed after the first illumination of the wires. (b) Echographic signals observed after four iterations of the time-reversal process. N is the number of transducers in the array, there are N N inter-element impulse response functions. h lm (t) is the signal delivered by transducer number l when a temporal delta function δ(t) is applied to the transducer number m. These N 2 signals provide a complete description of the system. Indeed, if e m (t), 1 m N, are the transmitted signals then the received signals are given by the equation e m (t) N r l (t) = k lm (t) t e m (t). (22) m=1

21 Topical review R21 1 Iteration Maximum of the pressure,5 Iteration 1 Iteration 2 Iteration 3 Iteration Distance from the array axis (mm) Figure 13. Pressure patterns measured in the wire plane after each iteration. This equation simplifies in the frequency domain in a matrix form R(ω) = K(ω)E(ω) (23) where E(ω) and R(ω) are the transmitted and the received vector signals of N components, and K(ω) is the N N transfer matrix of the system The iterative time-reversal process. The time-reversal process can now be described. The matrix relation between transmitted and received signals provides an expression of the transmitted signals at each iteration of the process. A time-reversal operation t tis equivalent to a phase conjugation in the frequency domain so that if E (ω) is the first transmitted signal of an iterative time-reversal process then the second transmitted signal is the phase conjugate of the received signals E 1 (ω) = K (ω)e (ω). After two time-reversal operations the transmitted signal depends linearly on the first transmitted one as follows: E 2 (ω) = K (ω)k(ω)e (ω). After 2n time-reversal operations the transmitted signal is E 2n (ω) = [K (ω)k(ω)] n E (ω). (24) K (ω)k(ω) is called the time-reversal operator. As a consequence of the well known reciprocity theorem, the inter-element impulse response from element m to element l is equal to the inter-element impulse response from element l to element m, so the transfer matrix is symmetrical. Due to the symmetry of K(ω), this operator is Hermitian with positive eigenvalues. Let V 1 (ω) be the projection of E (ω) on the eigenspace of highest eigenvalue λ 1 (ω). For large values of n the transmitted signal after 2n iterations of the time-reversal process is E 2n (ω) λ 1 (ω) n V 1 (ω). (25)

22 R22 Topical review In practice, at each iteration of the TR process, the coefficient λ 1 (ω) is compensated by some amplification, so the signal remains finite. We see therefore that the iterative TR process is convergent and the limit is V 1 (ω). The convergence is geometrical, depending on the ratio of the highest and the second highest eigenvalue λ 1 (ω)/λ 2 (ω). We need now to give a physical interpretation of these mathematical results and this can be done in the case of point-like well resolved scatterers The transfer matrix for point-like scatterers. In the case of point-like scatterers the transfer matrix can derived easily. We assume that the medium contains d point-like (Rayleigh) scatterers with complex frequency-dependent reflectivity coefficients C 1 (ω), C 2 (ω),...,c d (ω). Then the transfer matrix can be written as the product of three matrices: (1) a forward propagation matrix that describes the transmission and the propagation from the transducers to the scatterers, (2) a scattering matrix which is diagonal in the case of a single scattering process and (3) the backward propagation matrix. Let h il (t) be the diffraction impulse response function of the transducer number l to the scatterer number i with Fourier transform H il (ω). Let a e (t) and a r (t) be the transducer acoustoelectrical response in emission and in reception, with Fourier transforms A e (ω) and A r (ω). If the input signal at each element l is e l (t), then the pressure at the scatterer i is N p i (t) = h il (t) t a e (t) t e l (t). (26) l=1 This equation is written in the frequency domain as N P i = A e H il E l. l=1 The expression is simplified using a matrix notation: P = A e HE where E is the input vector signal, P is the vector representing the pressure received by the d scatterers and H is a matrix of dimensions N d called the diffraction matrix. In the case of single scattering, the pressure reflected by scatterer number i is C i P i. Therefore, the vector of reflected pressures is CP where C is a diagonal matrix of coefficients C ij = δ ij C i for all i, j 1,...,d. According to the reciprocity principle, the propagation from the scatterer number i to the transducer number l is also h il (t), so the back-propagation matrix is the transpose of the propagation matrix H. Consequently, the output signal R is R = A e A t r HCHE. The expression of the transfer matrix is K = A e A t r HCH Convergence of the iteration for well resolved scatterers. If the scatterers are ideally resolved, then the lines H i = (H i1,h i2,...,h in ) of the matrix H are orthogonal and the matrix A = C t H H is diagonal with coefficients N α i = C i H il 2 1 i D. (27) l=1 α i is called the apparent reflectivity of scatterer i.

23 Topical review R23 Then the operator describing 2n iterations of the time-reversal process is [K (ω)k(ω)] n = t H (A A) n H. If the first scatterer has the greatest apparent reflectivity so that α 1 > α 2,..., α D, then for large values of n we have [K (ω)k(ω)] n = α 1 2nt H1 H 1. If V is the first transmitted vector signal, then the limit of the iterative process is V 2n = V, H 1 α 1 2nt H1. (28) It is proportional to the vector t H1, which is the vector that would be emitted in a time-reversal process if the first scatterer were alone. Note that if V is perpendicular to this vector then the iteration will never reach focusing on the stronger scatterer. Fortunately, this situation is seldom satisfied in experiments Inverse scattering analysis and target resonance Beside their usefulness for the detection of reflecting objects, TRMs are also new means to carry out an inverse scattering analysis on a spatially extended elastic target, in other words to characterize a scatterer from the knowledge of its scattered field. The same analysis allows us also to develop a technique to put an elastic target in resonance. The interaction of a short ultrasonic beam with an extended solid target produces a complicated back-scattered field. A first reflected wave, the specular echo, that would be obtained if the target were perfectly rigid, is determined by the target geometry. Then it is followed by a series of waves, the elastic echo, which corresponds to the propagation of surface and volume waves around and inside the scatterer. These waves are generated at particular points on the target. They propagate at the surface or in the solid and they radiate into the fluid from different mode-conversion points on the scatterer which behave as secondary sources. Thomas et al [29] have studied the case of a hollow target where the elastic part is mainly due to circumferential waves (dispersive Lamb waves): the first symmetrical and antisymmetrical Lamb waves S and A. As these two waves are of different velocities, they can be separated experimentally by time windowing and their generation points on the target are located at different positions in accordance with Snell laws. Selecting and time reversing each wavefront separately, the time-reversed wave energy only concentrates on the generation points of the selected wave. This process enhances the generation of each specific Lamb wave compared to the other reflected waves. Iterating this process, we can build a waveform that is spatio-temporally matched to each of the vibration modes of the target Circumferential waves and time reversal. In the case of a thin air-filled hollow cylinder, the elastic part of the scattered field is due to circumferential waves similar to those propagating in a plate: the first symmetrical Lamb wave S and the first antisymmetrical Lamb wave A. These waves are generated at particular points on the scatterer. They propagate in the solid and they radiate into the fluid from different points on the scatterer, which behave as secondary sources. The nature of these waves can be determined from the knowledge of their radiation points (figure 14). As these two waves have different group velocities, they are easily separated experimentally by time windowing. These waves are generated at a given angle of incidence θ with respect to the normal to the surface. This angle satisfies the relation sin(θ) = C C φ where C is the sound velocity in water and C φ is the phase velocity of the Lamb wave. While propagating around the cylinder, it radiates into the fluid at the opposite angle with respect to the normal to the surface.

24 R24 Topical review θ θ A air water B θ Figure 14. Generation and radiation of a Lamb wave on a thin hollow cylinder. For an incident plane wave, two Lamb waves are generated at points A and B, symmetrical with respect to the incident direction. These two waves radiate backward from the same points A and B. The distance d AB between these points is given by d AB = D C (29) C φ where D is the diameter of the cylinder Experimental results. Experiments have been conducted with a linear array of 128 transducers of central frequency 3.5 MHz on a steel cylinder of diameter 2 mm and thickness approximately.6 mm. The cylinder axis is perpendicular to the array of transducers and is located at a depth of 8 mm (figure 15). A short pulse is launched by the centre element of the array. The signal scattered by the cylinder is recorded on the whole array. The signal received by the centre element is composed first of the strong specular echo and then of the elastic echo (figure 16). On the elastic part, two pairs of wavefronts with interference fringes can be seen. These wavefronts correspond to the different circumferential waves that have turned once around the shell. The one coming first is supposed to be the S Lamb wave; the second is the A Lamb wave. The symmetry of the fringes with respect to the array axis reveals the two symmetrical point-like sources. Each pair of sources must be associated with one surface wave. To confirm this observation, a time window selects the first echo in the elastic part of the signal. This echo is time reversed and the time-reversed field is scanned with a needle hydrophone along a line perpendicular to the cylinder axis. At each location of the hydrophone, the time dependence of the pressure is recorded. The directivity pattern which represents the maximum of the pressure field (figure 17) shows two symmetrical lobes. The distance between them corresponds to a phase velocity of 5.1 mmµs 1, which confirms that it corresponds to the S Lamb wave. The same process is applied to the second part of the elastic echo. In this case, the distance between the lobes is 11 mm; this corresponds to a phase velocity equal to 2.3mmµs 1, which is the velocity of the A Lamb wave. Time-reversal processing can also be applied to generate selectively one of the circumferential waves. In this case the selected time-reversed field is transmitted without removing the cylinder. The time-reversed wave is spatially adapted to the chosen circumferential wave; its energy is focused on its two generation points. The enhancement of

25 Topical review R25 y x z Figure 15. Experimental set-up. Figure 16. Echo of the shell received by the 128 transducers after transmission of a short pulse by the centre element of the array. The first wavefront is the specular echo, the second is the contribution of the S Lamb wave and the third the contribution of the A Lamb wave. Maximum of the pressure 1,2,8, ,5 7,5 S A Distance from the axis of the shell (mm) Figure 17. Pressure pattern measured in the plane of the shell after time reversal of the echo corresponding to the A and S Lamb waves. the corresponding wave can be seen on the new echo from the cylinder. The elastic part of the echo is dominated by the selected wave as shown in figure 18 for the wave A and in figure 19 for the wave S. The wave A is attenuated very fast (it has a high radiation coefficient), so only one turn around the shell can be seen, but the wave S propagates on several turns around

26 R26 Topical review Figure 18. Echo of the shell after time reversal of the A wavefront. Figure 19. Echo of the shell after time reversal of the S wavefront. the shell and the contribution of three turns can be seen in the echo. In the following section, we shall see that the DORT method allows us to go further into this analysis. 4. The DORT method The DORT method was derived from the theoretical study of iterative TRMs. From a theoretical point of view, this method can be considered as a generalization of iterative TRMs. It provides a new approach to active detection with arrays of transducers. DORT is the French acronym for Décomposition de l Opérateur de Retournement Temporel (decomposition of the time-reversal operator). It is a technique of analysis of pulse-echo measurements that can be acquired with arrays of transmit receive transducers Principle The iterative time-reversal process allows a selective focusing on the most reflective of a set of scatterers. It is also interesting to learn how to focus on the other scatterers. For example, this occurs in lithotripsy when several kidney stones exist. The DORT method provides a solution to this question [26, 31, 32]. This analysis consists in determining the possible transmitted waveforms that are invariant under the time-reversal process. We show that, for a set of well resolved scatterers, the focusing on one of them is such an invariant.

27 Topical review R Eigenvectors of the time-reversal operator. As shown in the last section, if the scattering medium contains a number D of Rayleigh scatterers, the transfer matrix is the product of three matrices: K = t HCH. We say that the targets are perfectly resolved if the time-reversal focusing on one of them does not produce energy on the others. In that case, the eigenvectors of the time-reversal operator are known. They are the vectors t H i 1 <i<d complex conjugates of the lines of the matrix H associated with the eigenvalues ( N ) 2 λ i = C i 2 H il 2 1 <i<d. (3) l=1 The vector t Hi is the signal that would be transmitted after one time-reversal process if only the target number i were present. (The detailed proof can be found in [31, 32].) Consequently, for well resolved point-like scatterers the number of non-zero eigenvalues is equal to the number of scatterers. Furthermore, if the scatterers have different apparent reflectivities, each eigenvector is associated with one scatterer and its phase and amplitude are the ones to be applied on each transducer in the array in order to focus on that scatterer. Therefore the measurement and diagonalization of the time-reversal operator provides information on the number of targets and their positions. These results have motivated the implementation of the DORT method, which is performed in three steps. The first step is the measurement of the inter-element impulse responses of the system. Since the reception system operates in parallel, this measurement requires N transmit receive operations for an array of N transducers. The first transducer of the array is excited with a signal e(t) and the signals received on the N channels are stored. This operation is repeated for all the transducers of the array with the same transmitted signal e(t). The components of the transfer matrix K(ω) are obtained by a Fourier transform of each signal. This measurement could also be made with any multiplexed system by N 2 transmit receive operations. The second step is the diagonalization of the time-reversal operator K (ω)k(ω) at a chosen frequency. The eigenvalue distribution contains valuable information such as the number of secondary sources in the scattering medium. The third step is to back-propagate each eigenvector. This can be done either numerically or experimentally. From a mathematical point of view the diagonalization of the time-reversal operator is equivalent to the singular-value decomposition of the transfer matrix K. The eigenvalues of K (ω)k(ω) are the squares of the singular values of K. In the following we may present either the singular values or the eigenvalues. The DORT method shares some of the principles of eigenvector decomposition techniques that are used in passive source detection [33, 34]. However these last techniques assume statistically uncorrelated sources while the DORT method is active and deterministic, thus they should not be considered as competing techniques Selective focusing through an inhomogeneous medium Experimental set-up. The experimental set-up is similar to that presented in the last section to illustrate the efficiency of the iterative time-reversal process. The defocusing effect of the aberrator is illustrated by the pressure pattern obtained by cylindrical focusing on the

28 R28 Topical review Figure 2. Pressure pattern obtained by cylindrical focusing through the rubber layer. Cylindrical focusing through the aberrating layer. Maximum of the pressure field. 1,5 Off axis distance in mm. Figure 21. Echo of the two wires after illumination by the centre element of the array. two wires. These cylindrical laws were calculated to focus on the wires as if there were no aberrator; the resulting field is indeed not at all focused. The echo of the two wires (figure 21) contains two wavefronts whose amplitudes are very close. In this case 2 iterations of the time-reversal process would have been necessary to reach focusing on the dominant scatterer. Even in this case, we shall see that the DORT method allows us to separate the two echoes Results of the DORT method. The inter-element impulse responses were measured and the time-reversal operator was computed at the frequency of 3 MHz. The decomposition revealed two significant singular values (figure 22). Two sets of acoustic signals were built from the phase law of the corresponding eigenvectors (figure 23) and they were transmitted through the medium. The thus-produced pressure field was measured with a hydrophone in the plane of the wires. Each eigenvector focuses on one of the wires (figure 24). The quality of the focusing is similar to that expected in plain water.

29 Topical review R Figure 22. Singular values of the transfer matrix. 4 phase of V phase of V Figure 23. Phase law of the first and second eigenvectors. Maximum of the pressure field. 1 V1 V2, Off axis distance in mm. Figure 24. Pressure patterns measured at the position of the wires after transmission of the first and second eigenvectors Application to an inverse problem The preceding experiment shows the effectiveness of the DORT method to focus selectively on different scatterers through an inhomogeneous medium, but more generally this method isolates and classifies the scattering centres or secondary sources in the medium and can be used to analyse the scattering from extended objects. Consequently this method applies to the problem of a thin hollow cylinder previously treated with the TRM [35].

30 R3 Topical review First and second eigenvectors Third and fourth eigenvectors Fifth and sixth eigenvectors transducer number Figure 25. Modulus of the components of the six eigenvectors (from top to bottom 1 2, 3 4, 5 6) Circumferential waves and invariants of the time-reversal process. As we mentioned in the last section (figure 14), the two circumferential waves generated in A and B are linked by reciprocity. The first one can be obtained by time reversal of the second one. Both waves are invariant under two successive time-reversal processes; consequently they should be associated with eigenvectors of the time-reversal operator. In fact, due to the symmetry of the problem, they are both associated with the same two eigenvectors: one corresponding to the generation in phase of the two waves and the other to the generation in opposite phase. The cylinder and the transducer array used for this experiment are similar to those used in the last section except that only 96 transducers were used. On the elastic part of the signals displayed in figure 16, the two pairs of wavefronts correspond to the S and A Lamb modes. In fact, at this frequency, three Lamb waves can be generated: A,A 1 and S, but the wave A 1 is so dispersive in this frequency range that its contribution to the scattered field interferes with the contribution of A and S and can hardly be distinguished. The inter-element impulse responses have been measured and only the elastic part of the signal was used to compute the time-reversal operator. At 3.5 MHz, the diagonalization of the time-reversal operator shows the six greatest eigenvalues. The modulus of the components of each eigenvector (1 6) is represented versus array element (figure 25). The interference fringes are easily observed. They are the equivalent at one frequency of the interference pattern observed on the echoes. This means that an eigenvector corresponds to the interference of two coherent point sources. The numerical back-propagation of each eigenvector confirms this analysis and allows us to determine the distance between the sources (figure 26). Each pair of sources corresponds to

31 Topical review R31 S A1 A Off axis distance in mm Figure 26. Pressure patterns obtained by numerical propagation of eigenvectors 1, 3 and 5. Phase velocity (mm/µs) S exp A exp A1 exp A1 theo S theo A theo 2,4 2,8 3,2 3,6 4 Frequency (MHz) Figure 27. Dispersion curves: theory and experiment. one particular wave. At this frequency, the first and second eigenvectors are associated with the wave S, the third and fourth with the wave A 1 and the fifth and sixth with the wave A. The same calculation is performed at several frequencies from 2.2 to 4 MHz so that the dispersion curves for the three waves can be plotted (figure 27). These curves are very close to the theoretical curves obtained for a steel plate of thickness.6 mm. In particular, the determination of the cut-off frequency of the wave A 1 allows us to tell the thickness of the shell Spectrum of the time-reversal operator and resonance frequencies of the shell. The eigenvalue associated with one particular wave depends on the frequency; it is proportional to the level of the contribution of the wave to the scattered field. Beside the responses of the transducers, the generation and reradiation coefficients of the wave are responsible for these variations. Moreover if the dynamic and duration of the recorded signals allow us to detect several turns of the wave around the shell, a fast modulation of the corresponding eigenvalue is induced, the maxima corresponding to the resonance frequencies of the shell. In the experiment, the wave A is attenuated so fast that only one turn can be observed, but several turns of A 1 and S contribute to the scattered field. The six first eigenvalues of the time-reversal operator are represented versus frequency in figure 28. The two curves λ 1 (ω)

32 R32 Topical review λ1 λ2 λ3 λ4 λ5 λ6 2,2 2,4 2,6 2,8 3 3,2 3,4 3,6 Frequency (MHz) Figure 28. Eigenvalues of the time-reversal operator versus frequency. and λ 2 (ω) correspond to the wave S. Their maxima occur at the resonance frequencies of the shell corresponding to this wave. The width of the peaks is mainly due to the length and dynamic of the recorded signals, which allow us to see only three turns of the S wave around the shell. Similar observations can be made for the wave A 1. This wave is associated with the eigenvalues λ 3 (ω) and λ 4 (ω) around its resonance frequencies and with λ 5 (ω) and λ 6 (ω) near its anti-resonance frequencies. The DORT method provides the resonance s frequencies due to the waves S and A 1, with the great advantage that close resonance frequencies can be distinguished Highly resolved detection and selective focusing in a waveguide The problem of optimum signal transmission and source location in a waveguide has been the subject of many theoretical and experimental works. The propagation of an acoustic pulse inside a waveguide is a complex phenomenon. However, as described in section 2.2.1, several studies have shown how to take advantage of this complexity [4,13 18]. Time-reversal experiments inside a water waveguide using a TRM demonstrated how to refocus an incident acoustic field back to its source and to achieve high temporal and spatial compression. In these experiments, only transmission from sources to receivers was considered. In echographic mode, the signal reflected from a scatterer is extremely complex for it has travelled a double path through the waveguide. This renders the detection process very difficult as the Green function that is used in matched field processing is non-trivial and its calculation requires accurate knowledge of the medium. The natural question that arises is how to take advantage of the super-focusing property of time reversal to improve detection and separation of scatterers. We shall now see through a simple experiment that the DORT method applies well to this problem [35].

33 Topical review R33 array of 6 transducers 35 mm 2mm wires 4 mm Figure 29. Geometry of the experiment. Amplitude ( a. u. ) 2,E-3 1,5E-3 1,E-3 5,E-4,E+ -5,E-4-1,E-3-1,5E-3-2,E Time ( µs ) Figure 3. Typical echo of the wires: inter-element impulse responses k 284 (t) Experiment. The experiment is performed in a two-dimensional water waveguide, bounded by two water/steel plane interfaces. The water layer is 35 mm thick. The array consists of 6 transducers with a central frequency of 1.5 MHz; it spans the whole height of the guide. The array pitch is equal to.58 mm. The scatterers are two wires of diameters.1 and.2 mm, separated by 2 mm and placed perpendicular to the array axis at a distance of 4 mm (figure 29). As the average wavelength is 1 mm, both wires behave almost like point scatterers. For this range and this frequency, the free space diffraction focal width is 12 mm, so the two wires are not resolved by the system. The echographic signals recorded after a pulse is applied to one transducer of the array are very complex, with low signal to noise ratio. The inter-element response k 28 4 (t) is a typical example (figure 3). After approximately five reflections at the interfaces the signal can no longer be distinguished from noise. The echoes of the two wires are superimposed and cannot be separated in a simple manner. The 6 6 impulse response functions are measured and the transfer matrix is calculated at frequency 1.5 MHz. The decomposition reveals two singular values that are separated from the 58 noise singular values (figure 31). The noise singular values are partly explained by electronic and quantification noises. However, different second-order acoustical phenomena that are not taken into account in the model probably contribute to these singular values: among them, the defects of the interfaces, the elastic responses of the wires, the multiple echoes between the wires and also coupling between the transducers. The eigenvectors V 1 and V 2 have a complicated phase and amplitude distribution and it is impossible to tell to which scatterer each of them corresponds. These distributions are applied to the array of transducers. Namely, if V 1 = (A 1 e iϕ 1,A 2 e iϕ 2,...,A n e iϕ n ) is the first eigenvector, then the signal s p (t) = A p cos(ωt ϕ p ) is applied to transducer p. The thusproduced pressure field is measured across the guide at the range of the wires (figure 32). For each eigenvector, the wave is focused at the position of one wire. In both cases the residual level is lower than 18 db and the 6 db focal width is 1.4 mm. For comparison, the same experiment is achieved after removing the guide. In this case the wires are not resolved and only the first singular value is significant. The pressure pattern is measured for transmission of the first eigenvector; the focal width is 13 mm. Consequently, the guide allows us to achieve a focusing almost ten times finer than in free space. The angular

34 R34 Topical review 6 Singular values ( a. u. ) wires 1 "noise" singular values Figure 31. Singular values of the transfer matrix calculated at 1.5 MHz. Depth ( mm ) Normalized amplitude ( db ) V1 V2 no guide Figure 32. Pressure pattern measured across the guide at the range of the wires after transmission of the eigenvectors. First (solid) and second (lower dotted) eigenvectors obtained with the guide, and first (upper dotted) eigenvector without the guide. directivity of each transducer limits the number of reflections at the guide interfaces that can be recorded. This induces an apodization of the virtual array made up of the set of images of the real one. Taking this phenomenon into account, the focal width roughly corresponds to a virtual aperture consisting of eight pairs of images of the array DORT in the time domain In the five preceding sections, the analysis of the transfer function is performed frequency by frequency. Only a small part of the information contained in the inter-element impulse

35 Topical review R35 4 3,5 Eigenvalues (a.u.) 3 2,5 2 1,5 1 Frequency (MHz),5,8 1 1,2 1,4 1,6 1,8 2 2,2 Figure 33. Singular values of the transfer matrix versus frequency. response functions has been used. In fact, the decomposition of the time-reversal operator can be performed at any frequency. In order to obtain temporal signals, it would be natural to calculate the eigenvectors in the whole band of the transducers and to perform an inverse Fourier transform of the eigenvector function of frequency. However, this operation is nontrivial. The main reason is that the scatterers reflectivity generally depends on frequency, so that at one frequency the first eigenvector can be associated with one scatterer while it is associated with another one at another frequency. However, if the strengths of the scatterers are different enough then the first eigenvector may correspond to the same scatterer in the whole frequency band of the transducers. In this case, it is possible to build temporal signals from the eigenvectors. If the first eigenvector corresponds to one point-like scatterer, then the temporal signal will provide the impulse Green function connecting the scatterer with the array Construction of the temporal Green functions. The above-mentioned conditions are satisfied in the following example. The array of transducers and the waveguide are the same as in section 1. The range of the scatterers is 4 mm, the distance between the scatterers is 2 mm and their reflectivities differ by a factor of three in the frequency band of the transducers. The singular-value decomposition of the transfer matrix is calculated at each frequency of the discrete spectrum from.8 to 2.2 MHz. The singular-value distribution versus frequency is shown in figure 33: two singular values are away from the 58 noise singular values and well separated from each other. The impulse response function from the strong scatterer to the array can be reconstructed from the eigenvectors λ 1 (ω)v 1 (ω). Assuming the reflectivity of the scatterer is independent of the frequency, this response is the temporal Green function connecting the scatterer to the array convoluted by the acousto-electrical response of the transducer (figure 34, top). The same procedure applied to λ 2 (ω)v 2 (ω) provides the impulse Green function from the second scatterer to the array (figure 34, bottom). This result is of particular interest in a

36 R36 Topical review Figure 34. Signals reconstructed from the first eigenvector (a) and from the second eigenvector (b). These signals correspond to the impulse response from each wire to the array Time ( µs ) Figure 35. Time domain compression: signal received at the position of the first wire after transmission of the first temporal eigenvector.